System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 50

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Question 1
1 / -0

Two objects, a ring and a spherical shell of same mass and radius are released from the top of two identical inclined plane. If they are rolling without slipping, then ratio of speed of center of mass of the two objects when they will reach the bottom of the inclined plane is

A
$$\sqrt { 5 } : \sqrt { 3 }$$

B
$$\sqrt { 5 } : \sqrt { 6 }$$

C
$$\sqrt { 5 } : \sqrt { 2 }$$

D
$$\sqrt { 5 } : 1$$

Solution
Question 2
1 / -0

A solid sphere is thrown on a horizontal rough surface with initial velocity of centre of mass u without rolling. Velocity of its centre of mass when it starts pure rolling is

A
$$\frac{{3u}}{5}$$

B
$$\frac{{2u}}{5}$$

C
$$\frac{{5u}}{7}$$

D
$$\frac{{2u}}{7}$$

Solution
Question 3
1 / -0

A satellite is revolving around a planet in a circuit orbit of radius. $$'r'$$ and has angular momentum $$L$$. If it is taken into a higher orbit of radius $$'4r'$$, the charge in its angulatr momentum is

A
$$ L/4$$

B
$$L/2$$

C
$$3L/4$$

D
$$L$$

Solution

A satellite is revolving around a planet in a circuit orbit of
radius$$=r$$
angular momentum$$=L$$
If it is time taken into higher orbit of radius $$=4r$$
The charge in its angular momentum$$=$$
The relation between radius and angular momentum $$L\alpha\cfrac{1}{r}$$
$$\cfrac{r}{4r}=\cfrac{L_2}{4}\\L_2=\cfrac{1}{4}L_1=\cfrac{1}{4}L$$
Question 4
1 / -0

Line $$PQ$$ is parallel to y-axis and moment of inertia of a rigid body about $$PQ$$ line is given by $${\rm I} = 2{x^2} - 12x + 27$$ where $$x$$ is in meter and $${\rm I}$$ is in $$Kg - {m^2}$$ The minimum value of $${\rm I}$$ is :

A
$$27\,kg\,{m^2}$$

B
$$11\,kg\,{m^2}$$

C
$$17\,kg\,{m^2}$$

D
$$9\,kg\,{m^2}$$

Solution

We have,
$$I=2x^2-12x+27$$
$$\Rightarrow \dfrac{dI}{dx}=4x-12$$
Putting $$\dfrac{dI}{dx}=0$$
$$\Rightarrow 4x-12=0$$
$$x=3$$
$$\Rightarrow I_{min}=2(3)^2-12(3)+27$$
$$=18-36+27$$
$$=27-18$$
$$I_{min}=9$$kg-$$m^2$$.
Question 5
1 / -0

Length width and mass of a rectangular plate are l, b and m respectively. The radius of gyration about the axis passing through centre and perpendicular to the plane is -

A
$$\sqrt { \frac { { l }^{ 2 }+{ b }^{ 2 } }{ 2 } } $$

B
$$\sqrt { \frac { { l }^{ 2 }+{ b }^{ 2 } }{ 8 } } $$

C
$$\sqrt { \frac { { l }^{ 2 }+{ b }^{ 2 } }{ 10 } } $$

D
$$\sqrt { \frac { { l }^{ 3 }+{ b }^{ 3 } }{ 10 } } $$
Question 6
1 / -0

Moment of inertia of a thin rod of mass $$M$$ and length $$L$$ about an axis passing through its center is $$\dfrac{ML^2}{12}$$. Its moment of inertia about a parallel axis at a distance of $$\dfrac{L}{4}$$ from this axis is given by

A
$$\dfrac{ML^2}{48}$$

B
$$\dfrac{ML^3}{48}$$

C
$$\dfrac{ML^2}{12}$$

D
$$\dfrac{7ML^2}{48}$$

Solution

By parallel axis theorem
$$I_1=I_0+md^2$$
$$I_1=\cfrac{ML^2}{12}+M(\cfrac{L}{H})^2=\cfrac{4ML^2+3ML^2}{48}=\cfrac{7ML^2}{48}$$
Question 7
1 / -0

Four bodies of masses 1,2,3,4 kg respectively are placed at the corners of a square of side $$'a'$$. Coordinates of centre of mass are (take $$1\ kg$$ at origin, $$2\ kg$$ on X-axis and $$4\ kg$$ on Y-axis)

A
$$\Big \lgroup \dfrac{7a}{10}, \dfrac{a}{2} \Big \rgroup$$

B
$$\Big \lgroup \dfrac{a}{2}, \dfrac{7a}{10} \Big \rgroup$$

C
$$\Big \lgroup \dfrac{a}{2}, \dfrac{3a}{10} \Big \rgroup$$

D
$$\Big \lgroup \dfrac{7a}{10}, \dfrac{3a}{2} \Big \rgroup$$

Solution
Question 8
1 / -0

The center of mass of the dipole

A
Has no acceleration

B
Has acceleration with positive x and y components

C
Has acceleration with positive x component and negative y component

D
Has acceleration with negative x component and positive y component

Solution

The center of mass velocity is the sum of each mass's momentum divided by the total mass of the system. The acceleration of the center of mass is the sum of all of the products of each mass and its acceleration divided by the total mass: Since Σ F = ma, ,It has no acceleration
Question 9
1 / -0

A 2 kg body and a 3 kg body are moving along the x-axis. At a particular instant the 2 kg body has the velocity of 2m/s and 3kg body has 7m/s. The velocity of the centre of mass at that instant is :-

A
5 m/s

B
1 m/s

C
0

D
$$\frac { 12 }{ 5 } m/s$$

Solution
Question 10
1 / -0

$$16\ kg$$ and $$9\ kg$$ are separated by $$25\ cm$$. The velocity with which a body should be projected from the midpoint of the line joining the two masses so that it just escape is :

A
$$\sqrt{g}$$

B
$$\sqrt{2gR}$$

C
$$2\sqrt{G}$$

D
$$2\sqrt{g}$$

Solution